CN107272692A - Unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection - Google Patents

Unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection Download PDF

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CN107272692A
CN107272692A CN201710584720.0A CN201710584720A CN107272692A CN 107272692 A CN107272692 A CN 107272692A CN 201710584720 A CN201710584720 A CN 201710584720A CN 107272692 A CN107272692 A CN 107272692A
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mrow
mtd
mtr
mfrac
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夏元清
张金会
李春明
付梦印
李胜飞
翟弟华
柴森春
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Beijing Institute of Technology BIT
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0219—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory ensuring the processing of the whole working surface

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  • Aviation & Aerospace Engineering (AREA)
  • Radar, Positioning & Navigation (AREA)
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  • Other Investigation Or Analysis Of Materials By Electrical Means (AREA)

Abstract

The present invention proposes a kind of unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection, can improve tracking effect of the four-wheel steering pilotless automobile in high-speed passing.Comprise the following steps：Step one：Set up Three Degree Of Freedom four-wheel steering automobile single track Controlling model；Step 2：The Controlling model set up according to step one, it is theoretical according to differential flat, drive lacking is controlled model and is transformed to the input and output coupling model without zero dy namics subsystem with disturbance；Step 3：Path planning layer is set up on tracing control layer.Step 4：The input and output coupling model set up according to step 2, designs the automatic disturbance rejection controller based on broad sense Proportional integral observer, and realization is tracked to the track that step 3 is cooked up.

Description

Unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection
Technical field
The invention belongs to four-wheel steering automobile path planning and tracing control field, it is related to a kind of based on differential flat and oneself The unmanned vehicle path planning and tracking and controlling method of anti-interference.
Background technology
Four-wheel steering system (all-wheel control system) is the important component of vehicle handling stability, its feature be Front and back wheel is anti-phase during low speed rotates, and reduces radius of turn；With rotating during high speed, changing Lane is easier.To trailer reversing On the premise of stability and vehicle intellectualized requirement continuous improvement, four-wheel steering automobile is carried out to reference locus (path planning) The requirement of accurate trajectory planning and tracing control is more and more urgent.
Differential flat theory is an effective approach for solving the problems, such as drive lacking.For other method, differential is put down Smooth theory can both handle minimum phase system (static feedback linearisation), and non-minimum phase system can be handled again, and (dynamic is anti- Linearization), nonlinear system is equivalent to the system for being input to flat output of no zero dy namics subsystem.But differential is flat Smooth condition is very harsh for a nonlinear system.Therefore some rational Approximation Methods are generally used, it is considered to which its is near Like the differential flat attribute of system, then residual error is estimated and compensated as disturbance and indeterminate.
Do not ensure that automatic driving vehicle handles any track exactly for giving the tracing control under desired trajectory The problem of tracking, from the angle of trajectory planning, it is considered to trajectory planning layer, Ye Jiju are set up on Trajectory Tracking Control layer Portion's planning layer.Labor is carried out to lane-change process, for the Transfer Parameters between reduced programming layer and key-course, it is proposed that A kind of method based on 5 order polynomial curve matchings, the tracing point of overtaking other vehicles cooked up is fitted.Employ Multivariable Linear Automatic disturbance rejection controller and broad sense Proportional integral observer carry out decoupling disturbance rejection control, realize overtaking other vehicles of cooking up and refer to rail The tracing control of mark, and compared with PID control method.Model initial error it is larger and occur in disturb (small angle Degree omitted items), disturb (longitudinal windage with lateral fitful wind) outside in the case of carried out MATLAB emulation, simulation result explanation is designed Method realize good tracking effect, linear active disturbance rejection controller (using high-order extended state observer) and broad sense ratio The tracking effect of integrator is essentially identical, in the case where disturbance quantity is especially complex, and three ranks of broad sense Proportional integral observer are more Item formula Approximation effect may be not so good as high-order extended state observer.Exist in sensor measuring state variable signal and measure noise When, employ Nonlinear Tracking Differentiator and Robust generalized Proportional integral observer is filtered, realize preferable filter effect, imitate True result shows that Robust generalized Proportional integral observer has stronger filter action.When therefore, for different control objects, Both can be selected between anti-interference effect and filter effect.Before designed planning layer can be obtained according to sensor The information and vehicle self information of square vehicle cook up the local desired trajectory information for getting around front vehicles, then expect rail by local Mark information input tracing control layer, in overtaking process, it is possible to achieve preceding object car is avoided, while to global reference path Tracking.Simulation result shows that designed double-deck control system can be realized in environment of overtaking other vehicles to the reliable, steady of reference locus Fixed tracking.
The content of the invention
The present invention be directed to the defect of prior art, a kind of unmanned vehicle path rule based on differential flat and active disturbance rejection are proposed Draw and tracking and controlling method, four-wheel steering automobile can be improved to the tracking effect for the track cooked up.
The present invention is achieved through the following technical solutions：
A kind of unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection, comprise the following steps：
Step one：Set up Three Degree Of Freedom four-wheel steering automobile single track Controlling model；
Step 2：The Controlling model set up according to step one, it is theoretical according to differential flat, drive lacking is controlled model conversion For the input and output coupling model without zero dy namics subsystem with disturbance；
Step 3：From the angle of trajectory planning, labor, reduced programming layer and key-course are carried out to lane-change process Between Transfer Parameters, based on 5 order polynomial curve-fitting methods, the tracing point of overtaking other vehicles cooked up is fitted；
Step 4：The input and output coupling model set up according to step 2, designs linear automatic disturbance rejection controller and broad sense ratio The track of overtaking other vehicles that example integral observer is cooked up to step 3 is tracked, and it is micro- that described controller includes high order linear tracking Divide device, high-order extended state observer and linear Feedback Control rule, finally obtain actual controlled quentity controlled variable.
Beneficial effects of the present invention：
The present invention has carried out model conversion to drive lacking Three Degree Of Freedom four-wheel steering automobile with differential flat theory, non- Linearly, coupling, drive lacking model be transformed into no zero dy namics subsystem it is defeated to it is flat enter output model, and consider The inaccurate part of model and it is inside and outside disturb, indeterminate etc. is all classified as disturbances suffered by system, then in planned course Controller is designed on the basis of planning layer to be controlled.Differential flat system can be by interior lively state feedback equivalence in a line Sexual system (feedback linearization), passes through the reference locus of the good flat output of tracking, it is possible to tracked state variable defeated with system The reference locus gone out.
Brief description of the drawings
Fig. 1 four-wheel steering automobile single track models.
Embodiment
The invention will be described further below.
A kind of the unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection of the present invention, including with Lower step：
The first step：Four-wheel steering automobile Three Degree Of Freedom drive lacking single track Controlling model is set up, is described as follows：
Wherein, vx,vy, r be respectively four-wheel steering automobile longitudinally, laterally and yaw velocity, flfAnd flrIt is by starting Machine, brake torque and longitudinal force caused by friction, are given by：
Wherein, ε is longitudinal resultant force fmThe constant value distribution coefficient of longitudinal force in front and back wheel, its span be from 0 i.e. rear wheel drive truck is to 1 i.e. front-wheel-drive cars, it is contemplated herein that automobile takes ε=1 only by the situation of front-wheel drive；
fsfAnd fsrIt is the horizontal lateral deviation power of tire
Wherein
Wherein, αfAnd αrThe F in front and back wheel side drift angle, model is represented respectivelyx,FyAnd TzRepresent respectively X, Y direction it is outer Portion's power and torque disturbance, make the f in first equation of modelsf=fsr=0；The formula of above-mentioned power is brought into model, and carried out Small angle approximation, obtains following equations
Wherein, Fx′,Fy' and Tz' it is residual error after small angle approximation, take vx,vy, r is state variable, δfAnd fmFor control Variable, by the above-mentioned form for being modeled as state space
Wherein
G1(x,t)u1u2+ ψ (x, u, t) is disturbed or indeterminate inside and outside regarding as, and real system equation is changed into
Second step：The Controlling model of foundation in the first step, is become using differential flat theory to Controlling model Change, turn to Multivariable Coupling and be input to flat output model, the problem of solving drive lacking is specific as follows：
Approximation system
Flat be output as
State variable is expressed as by flat output and its limited order derivative
Wherein
k1k8+k2k7≠0
And have
So
Wherein
As long as matrixIt is reversible, then approximation system is exactly differential flat, i.e.,
The condition is easily met in practice, therefore real system is turned to：
Wherein,The item of had an impact system is included, including：It is inside and outside disturb, indeterminate, little Jiao The approximate residual error of degree, in addition to it is vertically and horizontally approximate in power formula after residual error.
3rd step：Trajectory planning is carried out to overtaking process, it is assumed that overtake other vehicles beginning when, vehicle P is with speed v along x-axis side To at the uniform velocity travelling, in its preceding object car O with speed v1At the uniform velocity equidirectional traveling, wherein 0 ＜ v1＜ v, vehicle is tried to cut in Obstacle car O, in whole overtaking process, total distance in X-direction is that total distance in D, Y direction (is equal in two tracks The distance between heart line) it is W, total passing time is T, and lane-change terminal abscissa is X；
Overtaking process can be subdivided into three phases：
First stage：Turned to from former track；
Second stage：In fast straight-line travelling；
Phase III：Former track is returned to from fast；
Phase III optimal trajectory is designed as：
Two constraintss：
(1) peak acceleration circle constraint：
(2) x-axis direction constraint of velocity：
Wherein μ is ground friction coefficient, and N is normal pressure, is known parameters.
S=vT-D is made, optimization problem can be obtained：
4th step：The Multivariable Coupling input/output model set up according to second step, is entered using LADRC controllers (GPI) Row control with the gained track of tracking step three, it is contemplated that the inside and outside complexity disturbed, and auto-disturbance rejection technology advantage, and then use Automatic disturbance rejection controller is controlled, and using linear active disturbance rejection controller (LADRC), controller design step is as follows：
1. Nonlinear Tracking Differentiator TD design：
The linearity tracking differentiator that designs herein single order higher than conventional design method, is to obtain the height of reference signal Single order
Derivative, to add feedforward term in control.The main function of Nonlinear Tracking Differentiator is the band measured to sensor Measure
The reference signal of noise is filtered.
2. high order linear extended state observer LESO design
ConsiderComplexity, employ high order linear extended state observer.
3. the design of Feedback Control Laws：
Actual controlled quentity controlled variable

Claims (9)

1. a kind of unmanned vehicle path planning and tracking and controlling method based on differential flat and active disturbance rejection, it is characterised in that including Following steps：
Step one：Set up Three Degree Of Freedom four-wheel steering automobile single track Controlling model；
Step 2：The Controlling model set up according to step one, it is theoretical according to differential flat, drive lacking is controlled model and is transformed to band There is the input and output coupling model without zero dy namics subsystem of disturbance；
Step 3：From the angle of trajectory planning, labor is carried out to lane-change process, between reduced programming layer and key-course Transfer Parameters, based on 5 order polynomial curve-fitting methods, the tracing point of overtaking other vehicles cooked up is fitted；
Step 4：The input and output coupling model set up according to step 2, designs linear automatic disturbance rejection controller and broad sense ratio product The track of overtaking other vehicles for dividing observer to cook up step 3 is tracked, and described controller includes high order linear and tracks differential Device, high-order extended state observer and linear Feedback Control rule, finally obtain actual controlled quentity controlled variable.
2. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 1 and tracing control side Method, it is characterised in that set up Controlling model using following methods in step one：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>rv</mi> <mi>y</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>cos&amp;delta;</mi> <mi>f</mi> </msub> <mo>-</mo> <msub> <mi>f</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>sin&amp;delta;</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>cos&amp;delta;</mi> <mi>r</mi> </msub> <mo>-</mo> <msub> <mi>f</mi> <mrow> <mi>s</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>sin&amp;delta;</mi> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>rv</mi> <mi>x</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>sin&amp;delta;</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>cos&amp;delta;</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>sin&amp;delta;</mi> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>s</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>cos&amp;delta;</mi> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>J</mi> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>l</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>sin&amp;delta;</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>cos&amp;delta;</mi> <mi>f</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>l</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>sin&amp;delta;</mi> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>s</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>cos&amp;delta;</mi> <mi>r</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>T</mi> <mi>z</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, m is car mass, vx,vy, r be respectively four-wheel steering automobile longitudinally, laterally and yaw velocity, δf(δr) be (rear) wheel steering angle, l before automobilef(lr) automobile barycenter CG with before (rear) wheel distance, Cf(Cr) be before (rear) Wheel slip it is firm Degree, flfAnd flrIt is the longitudinal force caused by engine, brake torque and friction, is given by：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;epsiv;f</mi> <mi>m</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mrow> <mi>l</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&amp;epsiv;</mi> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mi>m</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, ε is longitudinal resultant force fmThe constant value distribution coefficient of longitudinal force in front and back wheel, span is from 0 i.e. rear wheel driving Electrical automobile is front-wheel-drive cars, f to 1sfAnd fsrIt is the horizontal lateral deviation power of tire, Fx,FyAnd TzExternal force and torque are represented respectively Disturbance, it is contemplated that lateral deviation power fsfAnd fsrDriving/braking act on relative to longitudinal drive/brake force flfAnd flrDriving/braking make The very little for, therefore make the f in first equationsf=fsr=0, front and back wheel corner is controlled using simple ratio, i.e. δr=k δf, k is front and back wheel corner proportional control factor, obtains following equations
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>rv</mi> <mi>y</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mi>x</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>rv</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;delta;</mi> <mi>f</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>v</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>l</mi> <mi>f</mi> </msub> <mi>r</mi> </mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k&amp;delta;</mi> <mi>f</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>v</mi> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>l</mi> <mi>r</mi> </msub> <mi>r</mi> </mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mi>y</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>J</mi> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>l</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>(</mo> <mrow> <msub> <mi>&amp;delta;</mi> <mi>f</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>v</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>l</mi> <mi>f</mi> </msub> <mi>r</mi> </mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> </mfrac> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>l</mi> <mi>r</mi> </msub> <msub> <mi>C</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k&amp;delta;</mi> <mi>f</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>v</mi> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>l</mi> <mi>r</mi> </msub> <mi>r</mi> </mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>T</mi> <mi>z</mi> </msub> <mo>+</mo> <msubsup> <mi>T</mi> <mi>z</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, k is Fx′,Fy' and Tz' it is residual error after small angle approximation, take vx,vy, r is state variable, δfAnd fmBecome for control Amount, by the above-mentioned form for being modeled as state space
<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>u</mi> <mo>+</mo> <msub> <mi>g</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>&amp;psi;</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> 1
Wherein
<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>C</mi> <mi>f</mi> </msub> <mi>m</mi> </mfrac> <mfrac> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mi>f</mi> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mo>-</mo> <mfrac> <msub> <mi>C</mi> <mi>r</mi> </msub> <mi>m</mi> </mfrac> <mfrac> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mi>r</mi> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mi>f</mi> </msub> <msub> <mi>C</mi> <mi>f</mi> </msub> </mrow> <mi>J</mi> </mfrac> <mfrac> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mi>f</mi> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mi>r</mi> </msub> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> <mi>J</mi> </mfrac> <mfrac> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mi>r</mi> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>kC</mi> <mi>r</mi> </msub> </mrow> <mi>m</mi> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>l</mi> <mi>f</mi> </msub> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>-</mo> <msub> <mi>kl</mi> <mi>r</mi> </msub> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> <mi>J</mi> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>g</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <msub> <mi>l</mi> <mi>f</mi> </msub> <mi>J</mi> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
<mrow> <mi>x</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>v</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>u</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>u</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow>
G1(x,t)u1u2+ ψ (x, u, t) is disturbed or indeterminate inside and outside regarding as, and real system equation is changed into
<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>&amp;xi;</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
3. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 2 and tracing control side Method, it is characterised in that the input without zero dy namics subsystem for being transformed to carry disturbance by the controlled model of drive lacking is defeated Go out coupling model and use following methods：
Line translation is entered to controlled model using differential flat is theoretical, input and output with multiple variable model is converted into：
Approximation system
<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>u</mi> </mrow>
Flat be output as
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mi>f</mi> </msub> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>-</mo> <msub> <mi>kl</mi> <mi>r</mi> </msub> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> <mi>J</mi> </mfrac> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>kC</mi> <mi>r</mi> </msub> </mrow> <mi>m</mi> </mfrac> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
State variable is expressed as by flat output and its limited order derivative
<mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>y</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>8</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>8</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>k</mi> <mn>7</mn> </msub> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>7</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>8</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>k</mi> <mn>7</mn> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </mrow>
Wherein
<mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mi>f</mi> </msub> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>-</mo> <msub> <mi>kl</mi> <mi>r</mi> </msub> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> <mi>J</mi> </mfrac> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>kC</mi> <mi>r</mi> </msub> </mrow> <mi>m</mi> </mfrac> <mo>,</mo> <msub> <mi>k</mi> <mn>3</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>C</mi> <mi>f</mi> </msub> </mrow> <mi>m</mi> </mfrac> <mo>,</mo> <msub> <mi>k</mi> <mn>4</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> <mi>m</mi> </mfrac> <mo>,</mo> </mrow>
k7=k3+k4+k5+k6,
<mrow> <msub> <mi>k</mi> <mn>8</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msubsup> <mi>y</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>5</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>l</mi> <mi>f</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>6</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>l</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msubsup> <mi>y</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>k</mi> <mn>9</mn> </msub> </mrow>
k1k8+k2k7≠0
And have
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>u</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>&amp;Lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow>
So
<mrow> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>&amp;Delta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mi>&amp;Lambda;</mi> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
Wherein
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;</mi> <mn>21</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>k</mi> <mn>7</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>9</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;</mi> <mn>22</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <msub> <mi>k</mi> <mn>1</mn> </msub> <mi>m</mi> </mfrac> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>k</mi> <mn>7</mn> </msub> <mi>m</mi> </mfrac> <mfrac> <msub> <mi>x</mi> <mn>2</mn> </msub> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mfrac> <mo>-</mo> <mfrac> <msub> <mi>k</mi> <mn>9</mn> </msub> <mi>m</mi> </mfrac> <mfrac> <msub> <mi>x</mi> <mn>3</mn> </msub> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Lambda;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Lambda;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>k</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mfrac> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mo>-</mo> <mfrac> <msub> <mi>x</mi> <mn>2</mn> </msub> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mfrac> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>k</mi> <mn>9</mn> </msub> <mrow> <mo>(</mo> <mfrac> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>3</mn> </msub> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mo>-</mo> <mfrac> <msub> <mi>x</mi> <mn>3</mn> </msub> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mfrac> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
As long as so matrixIt is reversible, then approximation system is exactly differential flat, i.e.,
<mrow> <mi>det</mi> <mrow> <mo>(</mo> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;</mi> <mn>11</mn> </msub> <msub> <mi>&amp;Delta;</mi> <mn>22</mn> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;</mi> <mn>12</mn> </msub> <msub> <mi>&amp;Delta;</mi> <mn>21</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <mrow> <mo>(</mo> <mo>-</mo> <msup> <msub> <mi>k</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>7</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>9</mn> </msub> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>&amp;NotEqual;</mo> <mn>0</mn> <mo>&amp;DoubleLeftRightArrow;</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>8</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>k</mi> <mn>7</mn> </msub> <mo>&amp;NotEqual;</mo> <mn>0</mn> </mrow>
The condition is easily met in practice, therefore real system can be turned to
Wherein,The item of had an impact system is included, including：It is inside and outside disturb, indeterminate, low-angle it is near As residual error.
4. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 3 and tracing control side Method, it is characterised in that further comprise in vertically and horizontally power formula it is approximate after residual error.
5. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 3 and tracing control side Method, it is characterised in that further, described constant value distribution coefficient ε=1, proportional control factor k=1.
6. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 3 and tracing control side Method, it is characterised in that in step 3 to overtaking process carry out trajectory planning, it is assumed that overtake other vehicles beginning when, vehicle P is with speed v At the uniform velocity travel along the x-axis direction, in its preceding object car O with speed v1At the uniform velocity equidirectional traveling, wherein 0 ＜ v1＜ v, vehicle Obstacle car O is tried to cut in, in whole overtaking process, total distance in X-direction is total i.e. two cars of distance in D, Y direction The distance between road center line is W, and total passing time is T, and lane-change terminal abscissa is X；Wherein overtaking process is divided into three Stage：
First stage：Turned to from former track；
Second stage：In fast straight-line travelling；
Phase III：Former track is returned to from fast；
Phase III optimal trajectory is designed as：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mi>v</mi> <mi>t</mi> <mo>+</mo> <mo>(</mo> <mi>v</mi> <mi>T</mi> <mo>-</mo> <mi>X</mi> <mo>)</mo> <mo>(</mo> <mo>-</mo> <mn>10</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>+</mo> <mn>15</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mn>4</mn> </msup> <mo>-</mo> <mn>6</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mn>5</mn> </msup> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mi>W</mi> <mo>+</mo> <mi>W</mi> <mo>(</mo> <mo>-</mo> <mn>10</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>+</mo> <mn>15</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mn>4</mn> </msup> <mo>-</mo> <mn>6</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> <mn>5</mn> </msup> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced>
Two constraintss：
(1) peak acceleration circle constraint：
<mrow> <mo>|</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>|</mo> <mo>&amp;le;</mo> <mfrac> <mrow> <mi>&amp;mu;</mi> <mi>N</mi> </mrow> <mi>m</mi> </mfrac> <mo>=</mo> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow>
(2) x-axis direction constraint of velocity：
<mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>&amp;le;</mo> <mi>t</mi> <mo>&amp;le;</mo> <mi>T</mi> </mrow>
S=vT-D is made, optimization problem can be obtained：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>M</mi> <mi>i</mi> <mi>n</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo>,</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <msup> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mfrac> <mn>10</mn> <mrow> <mn>7</mn> <mi>T</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <msup> <mi>S</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>W</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <mi>V</mi> <mi>S</mi> <mo>+</mo> <msup> <mi>V</mi> <mn>2</mn> </msup> <mi>T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mi>u</mi> <mi>b</mi> <mi>j</mi> <mi>e</mi> <mi>c</mi> <mi>t</mi> <mi> </mi> <mi>t</mi> <mi>o</mi> </mrow> </mtd> <mtd> <mrow> <mn>27</mn> <msup> <mi>A</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mn>4</mn> </msup> <mo>-</mo> <mn>256</mn> <msup> <mi>v</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo>&amp;le;</mo> <mn>900</mn> <msup> <mi>W</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>
7. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 3 and tracing control side Method, it is characterised in that further, described high order linear Nonlinear Tracking Differentiator is designed using following methods：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mi>hv</mi> <mn>12</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mn>12</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>12</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>r</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mn>11</mn> </msub> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> </mrow> <mo>)</mo> <mo>-</mo> <mn>2</mn> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>v</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>21</mn> </msub> <mo>+</mo> <msub> <mi>hv</mi> <mn>22</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>22</mn> </msub> <mo>+</mo> <msub> <mi>hv</mi> <mn>23</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mn>23</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>23</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>r</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mn>21</mn> </msub> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> </mrow> <mo>)</mo> <mo>-</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msub> <mi>v</mi> <mn>22</mn> </msub> <mo>-</mo> <mn>3</mn> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>v</mi> <mn>23</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
The reference signal that band for being measured to sensor measures noise is filtered；Wherein, v11(v21) flat defeated for tracking Go out reference signal y1r(y2r), v12(v22,v23) be used for obtaining reference signal y1r(y2r) first derivative, r1,r2It is tracking differential The velocity factor of device, h is sampling step length.
8. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 3 and tracing control side Method, it is characterised in that further, described high-order extended state observer is designed using following methods：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>11</mn> </msub> <mo>-</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>11</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>12</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>11</mn> </msub> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>12</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>12</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>13</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>12</mn> </msub> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>13</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>13</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>14</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>13</mn> </msub> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>14</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>14</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>b</mi> <mn>14</mn> </msub> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>21</mn> </msub> <mo>-</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>21</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>22</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>21</mn> </msub> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>22</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>23</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>22</mn> </msub> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>23</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>23</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>24</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>23</mn> </msub> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>24</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>24</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>25</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>24</mn> </msub> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>25</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>25</mn> </msub> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>b</mi> <mn>25</mn> </msub> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, e1,e2It is observation error, z1i(i=1,2,3,4), z2j(j=1,2,3,4,5) it is the defeated of extended state observer Go out, b1i(i=1,2,3,4), b2j(j=1,2,3,4,5) is the gain of extended state observer, U1,U2It is virtual controlling amount.
9. a kind of unmanned vehicle path planning based on differential flat and active disturbance rejection as claimed in claim 3 and tracing control side Method, it is characterised in that further, described linear Feedback Control rule is using following methods design：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>11</mn> </msub> <mo>-</mo> <msub> <mi>z</mi> <mn>11</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>U</mi> <mn>10</mn> </msub> <mo>=</mo> <msub> <mi>c</mi> <mn>11</mn> </msub> <msub> <mi>e</mi> <mn>11</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>21</mn> </msub> <mo>-</mo> <msub> <mi>z</mi> <mn>21</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>22</mn> </msub> <mo>-</mo> <msub> <mi>z</mi> <mn>22</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>U</mi> <mn>20</mn> </msub> <mo>=</mo> <msub> <mi>c</mi> <mn>21</mn> </msub> <msub> <mi>e</mi> <mn>21</mn> </msub> <mo>+</mo> <msub> <mi>c</mi> <mn>22</mn> </msub> <msub> <mi>e</mi> <mn>22</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>12</mn> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>10</mn> </msub> <mo>-</mo> <msub> <mi>z</mi> <mn>12</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>23</mn> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>20</mn> </msub> <mo>-</mo> <msub> <mi>z</mi> <mn>23</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, e11,e21,e22It is state error, U10,U20It is state error feedback rate control, c11,c21,c22It is error feedback control Coefficient processed；
Actual controlled quentity controlled variable
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>u</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mi>&amp;Delta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow> 5
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